Quantum-classical Liouville description

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

V. QUANTUM-CLASSICAL LIOUVILLE DESCRIPTION A. General Idea... 

M. Santer, U. Manthe, and G. Stock. Quantum-classical Liouville description of multidimensional nonadiabatic molecular dynamics. J. Chem. Phys., 114(5) 2001-2012, 2001. 

At present the practical applicability of the quantum-classical Liouville description and the semiclassical version of the mapping approach is limited because of the dynamical sign problem. As both methods have been proposed only in the last few years, however, they still hold a great potential for improvement. Finally it is noted that all methods considered in this review can in principle be interfaced with an on-the fly ah initio... 

The similar appearance of the quantum and classical Liouville equations has motivated several workers to construct a mixed quantum-classical Liouville (QCL) description [27 4]. Hereby a partial classical limit is performed for the heavy-particle dynamics, while a quantum-mechanical formulation is retained for the light particles. The quantities p(f) and H in the mixed QC formulation are then operators with respect to the electronic degrees of freedom, described by some basis states 4> ), and classical functions with respect to the nuclear degrees of freedom with coordinates x = x, and momenta p = pj — for example. 

We have presented some of the most recent developments in the computation and modeling of quantum phenomena in condensed phased systems in terms of the quantum-classical Liouville equation. In this approach we consider situations where the dynamics of the environment can be treated as if it were almost classical. This description introduces certain non-classical features into the dynamics, such as classical evolution on the mean of two adiabatic surfaces. Decoherence is naturally incorporated into the description of the dynamics. Although the theory involves several levels of approximation, QCL dynamics performs extremely well when compared to exact quantum calculations for some important benchmark tests such as the spin-boson system. Consequently, QCL dynamics is an accurate theory to explore the dynamics of many quantum condensed phase systems. 

We show how the quantum-classical evolution equations of motion can be obtained as an approximation to the full quantum evolution and point out some of the difficulties that arise because of the lack of a Lie algebraic structure. The computation of transport properties is discussed from two different perspectives. Transport coefficient formulas may be derived by starting from an approximate quantum-classical description of the system. Alternatively, the exact quantum transport coefficients may be taken as the starting point of the computation with quantum-classical approximations made only to the dynamics while retaining the full quantum equilibrium structure. The utility of quantum-classical Liouville methods is illustrated by considering the computation of the rate constants of quantum chemical reactions in the condensed phase. 

In these examples the dynamics is not confined to a single adiabatic potential energy surface so that the full quantum dynamics of the entire system must be followed in order to obtain the observable of interest. For large systems, typical of condensed phase applications, this is a computationally difficult, if not impossible, task. For this reason, we focus our attention on quantum-classical descriptions where such limitations are much less severe. In particular, the formulation based on the quantum-classical Liouville equation is the topic of the remainder of this chapter. 

The quantum-classical Liouville equation (QCLE) provides an approximate but accurate description of a quantum subsystem coupled in an arbitrary manner to a bath that can be described by classical dynamics in the absence of coupling to the quantum subsystem. The QCLE describes the time evolution of the partially Wigner transformed density matrix of the system p R,P,t) discussed above, and is given by ... 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

There is one final case, which we describe very briefly here and in more detail later. The classical description can be written in a form that is quite similar to the number operator representation in quantum mechanics. An operator 0 is assigned to molecule i, which is one if / is of type a and zero otherwise. Now, however, these operators do not themselves depend on the positions and momenta they follow a dynamics that is specified by the classical Liouville operator of the system. In particular, the Liouville operator determines the conditions under which species interconversion is possible. Hence, just as in the quantum mechanical case, the problem of the specification of the precise conditions for reaction is deferred to the Liouville operator. Section VI describes how such Liouville operators can be constructed. 

The solution of time evolution problems for classical systems is facilitated by introducing a classical phase space representation that plays a role in the description of classical systems in a matmer that is formally analogous to the role played by the coordinate and momentum representations in quantum mechanics. The state vectors T ) of this representation enumerate all of the accessible phase points. The phase function /(E ) is given by / (f ) = (f I/), which can be thought to represent a component of the vector f) in the classical phase space representation. The application of the classical Liouville operator (f ) to the phase function /(f ) is defined by (f )/(f ) = (f I/), where is an abstract op-... 

The classical Liouville equation does have an equivalent in quantum mechanics, which is needed for a consistent description of quantum statistical mechanics the quantum Liouville equation. Equilibrium quantum statistical mechanics requires the introduction of the density operator on an appropriate Hilbert space, and the quantum liouvUle equation for the density operator is a logical and necessary extension of the Schrodinger equation. The quantum Liouville equation can even be written, formally at least, in a form that resembles its classical counterpart. It allows for some weak and almost internally consistent form of dissipative dynamics, known as the Redfield theory, which finds its main use in relating NMR relaxation times to spectral densities arising from solvent fluctuations, although in recent... 

Therefore, decay to a final equilibrium state should be inherent in a proper dynamical description of a system. It was indicated at the end of Section 9.10 that in classical systems this decay is well described by the formalism of the Fokker- Planck equation, which itself was shown to be an extension of the classical Liouville equation. In view of the similarities between the classical and quantum Liouville equations, it seems a natural question to ask whether it is also possible to find an extension of the quantum Liouville equation that lets any initial density operator decay to the equilibrium density operator for that system. 

This contribution deals with the description of molecular systems electronically excited by light or by collisions, in terms of the statistical density operator. The advantage of using the density operator instead of the more usual wavefunction is that with the former it is possible to develop a consistent treatment of a many-atom system in contact with a medium (or bath), and of its dissipative dynamics. A fully classical calculation is usually suitable for a many-atom system in its ground electronic state, but is not acceptable when the system gets electronically excited, so that a quantum treatment must then be introduced initially. The quantum mechanical density operator (DOp) satisfies the Liouville-von Neumann (L-vN) equation [1-3], which involves the Hamiltonian operator of the whole system. When the system of interest, or object, is only part of the whole, the treatment can be based on the reduced density operator (RDOp) of the object, which satisfies a modified L-vN equation including dissipative rates [4-7]. 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

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